Motivic sheaves and intersection cohomology

Masaki Hanamura^{∗}

We propose a motivic refinement of a result in [BBFGK]. The formulation involves the notion of intersection Chow group, introduced by A. Corti and the author.

We are very grateful to the referee for useful suggestions.

1 Intersection Chow groups and lifting theorems

We consider quasi-projective varieties over k = ℂ. For a quasi-projective variety Z, CH_{s}(Z) denotes the Chow group of s-cycles on Z tensored with ℚ; if Z is smooth, CH^{r}(Z) := CH_{dim Z−r}(Z). We consider only constructible sheaves of ℚ-vector spaces. The singular (co-) homology, Borel-Moore homology, and intersection cohomology are those with ℚ-coefficients.

Relative canonical filtration.

The study of filtration on the Chow group of a smooth projective variety was started by Bloch and continued by several people; of most relevance to us now are the works of Beilinson, Murre and Shuji Saito. Beilinson explained the filtration in terms of the conjectural framework of mixed motives. Murre proposed a set of conjectures, Murre’s conjectures, on a decomposition of the diagonal class in the Chow ring of self-correspondences; he relates the decomposition to the filtration of Chow groups.

For X a smooth projective variety, its Chow group of codimension r cycles CH^{r}(X) should have a filtration F^{•} such that CH^{r}(X) = F^{0} CH^{r}(X), F^{1} CH^{r}(X) is the homologically trivial part, F^{2} CH^{r}(X) is perhaps the kernel of Abel-Jacobi map, and so on. The subquotient Gr_{F}^{ν} CH^{r}(X) should in some way be determined by cohomology H^{2r−ν}(X, ℚ).

A candidate for the filtration was proposed by S. Saito, see [Sa 1] [Sa 2]. We extend his definition as follows. If S = Spec k, it coincides with Saito’s filtration.

Let S be a quasi-projective variety, and X a smooth variety with a projective map p : X → S. For another smooth variety W with a projective map q : W → S, an element Γ ∈ CH_{dim X−s}(W ×_{S} X) induces a map Γ_{∗} : CH^{r−s}(W) → CH^{r}(X), see [CH]. The cycle class of Γ in Borel-Moore homology gives a map Γ_{∗} : Rq_{∗}ℚ_{W} [−2s] → Rp_{∗}ℚ_{X}; passing to perverse cohomology one has a map (for each ν)

pℋ^{2r−ν}Γ_{∗} : ^{p}ℋ^{2r−2s−ν}Rq_{∗}ℚ_{W} → ^{p}ℋ^{2r−ν}Rp_{∗}ℚ_{X} .

(Here ^{p}ℋ^{∗} stands for perverse cohomology.)

∗Mathematical Institute, Tohoku University, Aramaki, Aoba-ku, Sendai 980-8587, Japan.

We define a filtration F_{S}^{•} on CH^{r}(X) as follows. Let CH^{r}(X) = F_{S}^{− dim S} CH^{r}(X). Assume F_{S}^{ν} has been defined. Define

F_{S}^{ν+1} CH^{r}(X) := ∑ Image[Γ_{∗} : F_{S}^{ν} CH^{r−s}(W) → CH^{r}(X) ]

where the sum is over (q : W → S, Γ ∈ CH_{dim X−s}(W ×_{S} X) ) satisfying the following condition: the map ^{p}ℋ^{2r−ν}Γ_{∗} : ^{p}ℋ^{2r−2s−ν}Rq_{∗}ℚ_{W} → ^{p}ℋ^{2r−ν}Rp_{∗}ℚ_{X} is zero. One can show:

Proposition 1.1 The filtration F_{S}^{•} on CH^{r}(X) has the following properties.

(1) CH^{r}(X) = F_{S}^{− dim S} CH^{r}(X). For any Γ ∈ CH_{dim X−s}(W ×_{S} X), the induced map Γ_{∗} : CH^{r−s}(W) → CH^{r}(X) respects F_{S}^{•}.

(2) If ^{p}ℋ^{2r−ν}Γ_{∗} : ^{p}ℋ^{2r−2s−ν}Rq_{∗}ℚ_{W} → ^{p}ℋ^{2r−ν}Rp_{∗}ℚ_{X} is zero, then Γ_{∗} sends F_{S}^{ν} CH^{r−s}(W) to F_{S}^{ν+1} CH^{r}(X).

(3) The filtration is the smallest one with properties (1) and (2).

Intersection Chow group.

We refer to a forthcoming paper with A. Corti for details on intersection Chow groups. Let S be a quasi-projective variety, X a smooth variety, and p : X → S a projective map.

There is an algebraic Whitney stratification

S = S_{0} ⊃ S_{1} ⊃ ⋯ ⊃ S_{α} ⊃ ⋯ ⊃ S_{dim S}

of S, so that S_{α} − S_{α+1} is smooth of codimension α, satisfying the following condition. (i) p is smooth projective over S^{0} := S − S_{1}, and (ii) there is an algebraic stratification of X such that p is a stratified fiber bundle over each

stratum S_{α}^{0} := S_{α} − S_{α+1}. We then say p : X → S is a stratified map with respect to {S_{α}}. The stratification can be chosen to satisfy a stronger condition as follows.

Let X_{α} = p^{−1}S_{α}. There exist resolutions X̃_{α} → X_{α} (with X̃_{0} = X) such that

(i) the induced map p̃_{α} : X̃_{α} → S_{α} is smooth over S_{α}^{0}, and

(ii) there is a stratificaton on X̃_{α} such that p̃_{α} is a stratified fiber bundle over S_{β}^{0} for β ≥ α. (In other words, p̃_{α} is a stratified map with respect to {S_{β}}_{β≥α}.) In this case we say the data (p : X → S, {X̃_{α} → X_{α}}) is stratified with respect to {S_{α}}.

X̃_{α}
↙ ↓
X ↩ X_{α}
↓_{p} ↓
S ↩ S_{α}

Let ι_{α} : X̃_{α} → X be the induced map.

We now restrict ourselves to the birational case: let S be a quasi-projective variety and p : X → S a resolution of singularities. One has maps (d = dim S)

CH_{d−r}(X̃_{α})⟶^{ι_{α}_{∗}} CH^{r}(X)⟶^{ι_{α}^{∗}} CH^{r}(X̃_{α})

Each group has filtration F_{S}^{•}.

Define the intersection Chow group as a subquotient of the Chow group of X given by:

ICH^{r}(S) := \frac{∩_{α≥1}(ι_{α}^{∗})^{−1}F_{S}^{2r−d+1} CH^{r}(X̃_{α})}{∑_{α≥1} ι_{α}_{∗}F_{S}^{2r−d+1} CH_{d−r}(X̃_{α})}

Theorem 1.2 ICH^{r}(S) is well-defined (independent of the choice of stratification and resolution).

Denote by IH^{i}(S) the intersection cohomology with middle perversity and with ℚ-coefficients.

Proposition 1.3 There is a natural map

ICH^{r}(S) → IH^{2r}(S) .

The \underline{Conjectures}.

We recall three well-known conjectures concerning cohomology, Chow group, and higher Chow group of a smooth projective variety over a field. In this paper we refer to them as \underline{Conjectures}. The addition of the third conjecture is needed to prove the existence of a tstructure on the triangulated category of mixed motives. See [Ha].

1. Grothendieck’s Standard conjecture.

This concerns the functorial behavior of cycle classes in (singular or étale) cohomology. It has two components, the Lefschetz type conjecture and the Hodge type conjecture. For k = ℂ, the latter holds true (Hodge index theorem). The Lefschetz type conjecture itself consists of three statements, Conjecture (A), (B) and (C). Conjecture (C) says: the Künneth components of the diagonal class of a smooth projective variety are algebraic.

The standard conjecture implies the semi-simplicity of the category of pure homological motives (Grothendieck).

2. Murre’s conjecture (Bloch-Beilinson-Murre conjecture)

One of the formulation of the conjectural filtration on Chow group is due to Murre, and stated as the existence of a orthogonal decomposition to projectors of the diagonal class Δ_{X} in CH(X × X). To be precise, the conjecture states:

(A) Let X be a smooth projective variety. There exists a decomposition Δ_{X} = ∑ Π^{i} to orthogonal projectors in the Chow ring such that the cohomology class of Π^{i} is the Künneth component Δ(2 dim X − i, i). The decomposition is called the Chow-Künneth decomposition.

(B) Π^{i} with i = 0, ⋯, r − 1 or i = 2d, ⋯, 2r + 1 acts as zero on CH^{r}(X).

(C) Put F^{0} = CH^{r}(X), F^{1} = Ker Π^{2r}, F^{2} = Ker(Π^{2r−1}|F^{1}), ⋯, F^{r} = Ker(Π^{r+1}|F^{r−1}),F^{r+1} = 0. This is independent of the choice of the decomposition in (A).

(D) F^{1} = CH^{r}(X)_{hom}, the homologically trivial part.

Note a Chow-Künneth decomposition gives a decomposition in the category of Chow motives over k h(X) = ⨁ h^{i}(X), where h^{i}(X) carries cohomology in degree i only. For the category of Chow motives, see §2.

3. Variant of Beilinson-Soulé vanishing conjecture: Let (X, 0, P) be an object of the category of Chow motives CHℳ(k) whose realization is of cohomological degree ≥ 2r − n if n > 0 and > 2r if n = 0. Then P_{∗} CH^{r}(X, n) = 0.

When we give results that hold under the three \underline{Conjectures}, we will always say so; some of them require only the first two. For example, Proposition 1.4 (Under \underline{Conjectures} ) F_{S}^{ν} CH^{r}(X) = 0 for ν large enough.

We have:

Theorem 1.5 (Under \underline{Conjectures} ) The map p_{∗} : CH^{r}(X) → CH_{d−r}(S) induces a surjective map ICH^{r}(S) → CH_{d−r}(S).

Under \underline{Conjectures}, one has (1.5), which immediately implies the following theorem (1.6) in [BBFGK]. One has the cycle class map cl : CH_{d−r}(S) → H_{2(d−r)}^{BM}(S) (the latter is the Borel-Moore homology). There is a natural map IH^{2r}(S) → H_{2(d−r)}^{BM}(S).

Theorem 1.6 For any z ∈ CH_{d−r}(S), its class cl(z) ∈ H_{2(d−r)}^{BM}(S) can be (non-canonically) lifted to an element of intersection cohomology.

Indeed, we can show (1.6) without assuming \underline{Conjectures}, but still using the same ideas as for the proof of (1.5).

2 Motivic categories and decompositions of motives

Theory of Chow motives.

Let S be a quasi-projective variety over k = ℂ. Let CHℳ(S) be the pseudo-abelian category of Chow motives over S. It has the following properties (for details see [CH] ).

• An object of CHℳ(S) is of the form

(X, r, P) = (X/S, r, P)

where X is a smooth variety over k with a projective (not necessarily smooth) map p : X → S, r ∈ ℤ, and if X has connected components X_{i},

P ∈ ⨁_{i} CH_{dim X_{i}}(X ×_{S} X_{i})

such that P ∘ P = P. Here ∘ denotes composition of relative correspondences defined in [CH], which makes ⨁_{i} CH_{dim X_{i}}(X ×_{S} X_{i}) a ring with the diagonal Δ_{X} as the identity element. If (Y, s, Q) is another object, Y_{j} the components of Y, then

Hom((X, r, P), (Y, s, Q)) = Q ∘ (⨁_{j} CH_{dim Y_{j}−s+r}(X ×_{S} Y_{j})) ∘ P .

Composition of morphisms is induced from the composition of relative correspondences.

• Let h(X/S) = (X, 0, id) and h(X/S)(r) = (X, r, id). More generally, Tate twist is defined to be the functor (t ∈ ℤ)

K = (X, r, P) ↦ K(t) = (X, r + t, P)

on objects.

• One has a functor

CH^{t} : CHℳ(S) → V ect_{ℚ}, CH^{t}((X, r, P)) = P_{∗} CH^{r+t}(X) .

Note CH^{t}(K) = CH^{0}(K(t)) and CH^{r}(h(X/S)) = CH^{0}(h(X/S)(r)) = CH^{r}(X).

• If X and Y are smooth varieties with projective maps to S and f : X → Y a map over S, there corresponds a morphism

f^{∗} : h(Y/S) → h(X/S) .

If X, Y are equidimensional, there corresponds

f_{∗} : h(X/S) → h(Y/S)(dim Y − dim X) .

It is of use to define the homological motive of X/S: if X has components X_{i},

h′(X/S) := ⨁ h(X_{i}/S)(dim X_{i}) .

Then a map f : X → Y induces a morphism f_{∗} : h′(X/S) → h′(Y/S).

• Let D_{c}^{b}(S) = D_{c}^{b}(S, ℚ) be the derived category of sheaves of ℚ-vector spaces on S with constructible cohomology. There is the realization functor

ρ : CHℳ(S) → D_{c}^{b}(S)

such that on objects

(X, r, P) ↦ P_{∗}Rp_{∗}ℚ_{X}[2r] ,

(P_{∗} ∈ End_{D_{c}^{b}(S)}(Rp_{∗}ℚ_{X}) is a projector, and P_{∗}Rp_{∗}ℚ_{X} is its image, which exists since D_{c}^{b}(S) is pseudo-abelian.) Note ρ(h(X/S)(r)) = Rp_{∗}ℚ_{X}[2r] and

ρ(h′(X/S)(r)) = Rp_{∗}D_{X}[2r] ,

where D_{X} is the dualizing complex of X. Recall D_{X} = ℚ_{X}[2 dim X] if X is smooth.

Theory of Grothendieck motives.

We also have the pseudo-abelian category of Grothendieck motives over S. The main properties are the following.

Denote by P erv(S) be the abelian category of perverse sheaves of ℚ-vector spaces on S. There is a canonical full functor cano : CHℳ(S) → ℳ(S) and a faithful realization functor ρ : ℳ(S) → P erv(S). The following diagram commutes.

CHℳ(S) \underrightarrow{cano} ℳ(S)
↓_{ρ} ↓^{ρ}
D_{c}^{b}(S) ⟶^{^{p}ℋ^{∗}} P erv(S)

Here ^{p}ℋ^{∗} = ⨁_{i} ^{p}ℋ^{i} is the total perverse cohomology functor.

Relative decomposition of motives.

The following is in [CH] (for this, we only need the first two of the three Conjectures). This is a motivic analogue of the decomposition theorem for the total direct image in [BBD].

Theorem 2.1 (Under \underline{Conjectures}) Let p : X → S be as before. Let \{S_{α}\} be a Whitney stratification of S, and X̃_{α} → X_{α} resolutions such that (p : X → S, {X̃_{α} → X_{α}}) is stratified with respect to {S_{α}}. Then:

(1) There are local systems 𝒱_{α}^{j} on S_{α} − S_{α+1}, non-canonical direct sum decomposition in CHℳ(S)

h(X/S) = ⨁_{j,α} h_{α}^{j}(X/S)

and isomorphisms

ρ(h_{α}^{j}(X/S)) ≅ IC_{S_{α}}(𝒱_{α}^{j})[−j + dim S_{α}]

in D_{c}^{b}(S).

(2) For each i, the sum ⨁_{j≤i, α} h_{α}^{j}(X/S) is a well-defined subobject of h(X/S) (independent of the decomposition).

(3) The category ℳ(S) is semi-simple abelian, and the functor ρ : ℳ(S) → P erv(S) is exact and faithful.

Relative canonical filtration and motives.

For a projective map p : X → S with X smooth, the filtration on CH^{r}(X) can be interpreted in terms of motives as follows. Keeping the notation in the above theorem, define subobjects of h(X/S) by

pτ_{≤i}h(X/S) := ⨁_{j≤i, α} h_{α}^{j}(X/S)

(the sum over (j, α) with j ≤ i) and subquotients

pℋ^{i}h(X/S) := ⨁_{α} h_{α}^{i}(X/S) .

More generally for r ∈ ℤ, subobjects of h(X/S)(r)

pτ_{≤i}(h(X/S)(r) ) := ⨁_{j≤i+2r, α} h_{α}^{j}(X/S)(r)

and subquotients

pℋ^{i}(h(X/S)(r) ) := ⨁_{α} h_{α}^{i+2r}(X/S)(r)

are defined. Then we have

CH^{r}(X) = CH^{0}(h(X/S)(r)) = CH^{0}(⨁_{α,ν} h_{α}^{2r−ν}(X/S)(r) ) ,

F_{S}^{ν} CH^{r}(X) = CH^{0}(^{p}τ_{≤−ν}(h(X/S)(r)) ) = CH^{0}( ⨁_{μ≤2r−ν ,α} h_{α}^{μ}(X/S)(r) ) ,

and

Gr_{F_{S}}^{ν} CH^{r}(X) = CH^{0}(^{p}ℋ^{−ν}(h(X/S)(r)) ) = CH^{0}(⨁ h_{α}^{2r−ν}(X/S)(r) ) .

3 Outline of the proof of (1.5)

We start with a result on perverse cohomology. Let X be smooth, p : X → S a projective map, and assume (p : X → S, {X̃_{α} → X_{α}}) is stratified with respect to {S_{α}}. There are local systems 𝒱_{α}^{i} on S_{α}^{0} such that Rp_{∗}ℚ_{X} ≅ ⨁ IC_{S_{α}}(𝒱_{α}^{j})[−j + dim S_{α}]. Let d = dim X.

Proposition 3.1 (1) Let ι_{α}^{∗} : Rp_{∗}ℚ → i_{α}_{∗}Rp̃_{α}_{∗}ℚX̃_{α} be the natural map ι_{α} induces, and

pℋ^{i}(ι_{α}^{∗}) : ^{p}R^{i}p_{∗}ℚ → i_{α}_{∗}^{p}R^{i}p̃_{α}_{∗}ℚX̃_{α}

the induced map on perverse cohomology of degree i. The restriction to the direct summand IC_{S_{α}}(𝒱_{α}^{i})[dim S_{α}]

pℋ^{i}(ι_{α}^{∗}) : IC_{S_{α}}(𝒱_{α}^{i})[dim S_{α}] → i_{α}_{∗}^{p}R^{i}p̃_{α}_{∗}ℚX̃_{α}

is a split injection.

(2) Let ι_{α}_{∗} : ι_{α}_{∗}Rp̃_{α}_{∗}DX̃_{α}(−d)[−2d] → Rp_{∗}ℚ be the natural map, and

pℋ^{i}ι_{α}_{∗} : ι_{α}_{∗}^{p}ℋ^{i}Rp̃_{α}_{∗}DX̃_{α}(−d)[−2d] → ^{p}R^{i}p_{∗}ℚ

the induced map on perverse cohomology; here DX̃_{α} is the dualizing complex. This map factors through a split surjection

pℋ^{i}ι_{α}_{∗} : ι_{α}_{∗}^{p}ℋ^{i}Rp̃_{α}_{∗}DX̃_{α}(−d)[−2d] → IC_{S_{α}}(𝒱_{α}^{i})[dim S_{α}]

to the direct summand of the target.

We can extend the definition of the filtration F_{S}^{•} as follows. For any quasi-projective (possibly singular) variety Z with a quasi-projective map to S, one can define a filtration F_{S}^{•} on the Chow group CH_{s}(Z). This was done in [CH, §5] in the case S = Spec k, and the general case is similar. For a projective map of varieties over S, f : X → Y, the induced map f_{∗} : CH_{s}(X) → CH_{s}(Y) respects the filtrations F_{S}^{•}. If S → S′ is a closed immersion, and Z → S, then the filtrations F_{S}^{•} and F_{S′}^{•} on CH_{s}(Z) coincide.

For a quasi-projective variety T, viewing it as a variety over T, one has filtration F_{T}^{•} on CH_{s}(T). For this filtration, one has the following result. The proof uses the triangulated category of mixed motives over a base, the perverse t-structure on it, and the interpretation of the filtration F_{S}^{•} on CH_{s}(Z) in terms of the perverse truncation (similar to the interpretation in §2). See [Ha] for the case where the base is Spec k.

Lemma 3.2 (Under \underline{Conjectures}) For an irreducible quasi-projective variety T, F_{T}^{−2s+dim T +1} CH_{s}(T) = 0.

From now on we assume the \underline{Conjectures} throughout.

Let p : X → S be a desingularization. We have a decomposition h(X/S) = ⨁ h_{α}^{j}(X/S) as in (2.1). In this case h_{0}^{ν} = 0 for ν ≠ d, and it can be shown CH^{r}(h_{0}^{d}) = ICH^{r}(S) as a subquotient of CH^{r}(X).

Lemma (3.2) implies that p_{∗} : CH^{r}(X) → CH_{d−r}(S) passes to a map ICH^{r}(S) → CH_{d−r}(S). For the surjectivity we must show: For any a ∈ CH_{d−r}(S), there is an element b ∈ CH^{r}(X)

such that (i) p_{∗}(b) = a, and (ii) ι_{α}^{∗}(b) ∈ F_{S}^{2r−d+1} CH^{r}(X̃_{α}) for each α ≥ 1. Let a ∈ CH_{d−r}(S) and ν ≤ 2r − d + 1. Consider the following claim (I)_{ν}.

Claim (I)_{ν}.

(1) (Case ν ≤ 2r − d) there is an element b^{ν} ∈ CH^{r}(X) with (i) p_{∗}(b^{ν}) = a, and (ii) b^{ν} ∈ F_{S}^{ν} CH^{r}(X).

(2) (Case ν = 2r − d + 1) there is an element b^{2r−d+1} ∈ CH^{r}(X) satisfying the following (let b = b^{2r−d+1} for short): (i) p_{∗}(b) = a, and (ii) b ∈ F_{S}^{2r−d} CH^{r}(X) (not F_{S}^{2r−d+1} CH^{r}(X) !), and b mod F_{S}^{2r−d+1} ∈ Gr_{F_{S}}^{2r−d} CH^{r}(X) = ⨁_{α≥0} CH^{r}(h_{α}^{d}(X/S)) is contained in the first summand ICH^{r}(S) = CH^{r}(h_{0}^{d}(X/S)).

For ν small enough (I)_{ν} obviously holds: one can take any element satisfying (i). The larger ν is, the stronger (I)_{ν} is. What we must show is (I)_{2r−d+1}.

Proposition 3.3 Let ν ≤ 2r − d. We have (I)_{ν} ⇒ (I)_{ν+1}.

The proof of Proposition (3.3) is achieved by an argument that uses Proposition (3.1), the motivic interpretation of the filtration in §2, the two lemmas (3.2) and (3.4), and semi-simplicity of the category ℳ(S).

Lemma 3.4 If ν < 2r − 2 dim X̃_{α} + dim S_{α}, then h_{α}^{2r−ν}(X/S) is zero.

Indeed using (3.1) one shows the realization of h_{α}^{2r−ν}(X/S) is zero. Since ρ : ℳ(S) → P erv(S) is exact and faithful, it follows h_{α}^{2r−ν}(X/S) itself is zero.

References

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